What does "any later than" mean in the following examples?
Do not be any later than 11.30am.
I don't go there any later than 10 because the lines are too long.
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Although it's not totally clear from your question, I believe that what you're asking is what the word "any" is doing here. The main answer is, not much. The sentences mean almost the same thing if you delete the word "any".
However, "any" gives a little more openness (either as to permission, or possibility) to the arrival/event taking place earlier than the stated time. As written, with the word "any", your sentences, both imply that it would be okay (and maybe better) to show up before 11:30 am, or with the second sentence, that the speaker wants to go earlier than 10 if they can.
Without the word "any", they both have a little bit of a sense that the person will arrive, or should arrive, at their destination closer to the stated time.
"Any later than" is just another way of saying "no later than." Instead of saying, "Do not be any later than 11.30am," you could say, "Be here no later than 11:30 am" or "Be here before 11:30 am." Instead of "I don't go there any later than 10 because the lines are too long," you could say "I don't go there past 10 because the lines are too long."